We now turn to the perturbativity of the scalar potential. At the one-loop level, a large Higgs quartic coupling at EW scale may give
rise to a positive \(\beta _{\lambda _h}\)  function, which results in a monotonic increase of \(\lambda _{h}(\mu )\)  with \(\mu \)  and eventually develops a Landau pole at some scale \(\Lambda _{\rm L}\) .
However, the perturbativity of theory has already been invalid long before \(\mu \)  reaches \(\Lambda _{\rm L}\) .
At the two-loop level, \(\lambda _{h}(\mu )\)  may approach an ultraviolet fixed point where \(\beta _{\lambda _h}\rightarrow 0\)  and \(\lambda _{h}(\mu )\rightarrow \lambda _{\rm FP}\)  [1]}. For a given cutoff scale \(\Lambda \) , perturbation theory is expected to be reliable for the value of
\(\lambda _{\Lambda }\equiv \lambda _{h}(\Lambda )\)  in the range of \(\lambda _{\rm FP}/4-\lambda _{\rm FP}/2\)  [2]}.
Following Ref. [3]}, the approximate perturbativity bounds on the couplings are given by
\(\lambda _{h}(\mu )<\lambda _{\Lambda },~~ \lambda _{s}(\mu )<\lambda _{\Lambda },~~ \lambda _{m}(\mu )<\lambda _{\Lambda }, {~~\rm for~~} \mu <\Lambda .\) 
